Ricci flow and Perelman’s proof of the Poincare conjecture

Abstract

The Poincare conjecture was one of the most fundamental unsolved problems in mathematics for close to a century. This was solved in a series of highly original preprints by the Russian mathematician Grisha Perelman, for which he was awarded the Fields Medal (2006). Perelman’s proof, building on the work of Hamilton, was based on the Ricci flow, which resembles a nonlinear heat equation. Many of Perelman’s and Hamilton’s fundamental ideas may be of considerable significance in other settings. This article gives an exposition of the work, starting with some basic concepts.